Mathematics

· Analytic Geometry: Cartesian Plane and Reference Systems, Euclidean Distance and Pythagorean Theorem, The Equation of a Circle. The equation of a Parabola. The equation of a straight line.
· Three-Dimensional Euclidean Space: System of Coordinates, Euclidean Distance, Equation of a Sphere. Picture of a Paraboloid.
· Matrix and Matrix Algebra: Matrices and Vectors, System of Linear Equations, Matrix Addition, Matrix Multiplication, Rules for Matrix Multiplication.
· Functions of one Variable: Introductions, Domain and Range, Graph of Functions. Linear functions, Slope, Point-Slope and Point-Point formulas, Quadratic Functions, Concavity, intercepts with the axes, role of the discriminant (Delta) in drawing the parabola. Cubic Functions, Rational Functions, Power Functions. Exponentials, Logarithms and their properties. Intuitive definitions of limits to infinity and one-sided limits to vertical asymptotes.
· Limits: A Brief Introduction to Limits, more on the concept of Limit: left and right limits, limits to infinity, vertical asymptote.
· Differentiations: slope of curves, tangents and derivatives. Increasing and Decreasing Functions, Simple Rules for Differentiation, Sums, Products and
· Linear and Quadratic Approximation.
· Optimization: first and second derivative test, critical points, necessary and sufficient conditions, concave and convex functions, inflection points.
· Functions of two variables: concepts and first definitions. Domains. Partial derivatives, gradient and hessian matrix. First and second order Taylor approximations.
· More on Matrix algebra: determinant and trace for 2x2. Eigenvalues and eigenvectors for 2x2 matrices.
· Optimization for function of two variables: Local maxima and local minima for functions of two variables. Necessary conditions for optimality. Sufficient conditions for optimality. Examples and exercises.
· Constrained Optimization for function of two variables: Level curves, Equality constraints, Lagrange Multiplier Method, Multiple Solutions Candidates.